Scale-up of Bioprocesses
497
r =
-rjy
(11.20)
We will in the following simply call
rj
“viscosity”, but it is important not to confuse the dynamic
with the
kinematic viscosity.
The latter is the ratio between the dynamic viscosity and the density
of a fluid. The kinematic viscosity has the unit (m2 s'1).
Note 11.4. Shear stress as a tensor property
In many textbooks outside the field of fluid mechanics, the concept of stress is mathematically abused
much in the same way as done above. Stress should properly be treated as a tensor property {see e.g.
Mase, 1970). The state of stress in a material phase is fully described by the stress tensor, T (N m‘2). The
stress tensor includes both normal stress (i.e. pressure) and shear stress and can be written:
T = - p I - T
(1)
where p is the pressure (N m'2), I is the identity matrix and t is
the shear stress tensor
(N m'2). The stress
acting on a surface at any point, characterized by a normal vector,
n,
can be found from the stress tensor
by scalar multiplication according to:
K
= n T
+ * K
=-n-(/>I + T)
(2)
where tn is the state of stress. Note that tn has the same unit as T, i.e. N m'2, but is a vector instead of a
tensor. For a Newtonian incompressible fluid, the shear stress tensor can be written:
t =
- t/(V« +V«T)
(3)
where Vu is the velocity gradient (s'1), which can be said to be the tensor counterpart of the previously
defined shear rate,
y .
For
Newtonian fluids
the viscosity is independent of the shear rate -
i.e.
it is constant - whereas for
non-Newtonian fluids
it is a function of the shear rate. The viscosities of fluids vary over a wide
range, as seen in Table 11.6. Water, and many fermentation broths containing yeasts or bacteria, can
be considered to be Newtonian fluids. However, fermentations involving filamentous fungi, or
fermentations in which polymers are excreted, will often exhibit non-Newtonian behaviour.
There are several different kinds of non-Newtonian behavior. The most important type is the s
hear
rate dependent
viscosity. A fluid for which the viscosity decreases with increasing shear rate is
called a
pseudoplastic fluid.
The opposite,
i.e.
a fluid with an increasing viscosity with increasing
shear rate is called a
dilatant fluid.
Polymer solutions, or solutions containing filamentous fungi, are
often pseudoplastic, whereas e.g. rice straw in water or whipped cream exhibit dilatant behavior.
Certain materials do not flow until a treshold shear stress is exceeded. This is true for e.g. damp clay
or cheese. Although they are not in a strict sense fluids, they behave like fluids once the yields stress
is exceeded. These materials are called Bingham plastics or Bingham fluids.